**MAE 3181 Materials and Structures Laboratory**

**Laboratory Report #1**

**LVDT-Based Displacement Measurement Calibration**

**Summary**

This experiment is the process of calibrating a linear variable differential transducer. In this experiment we find the differential voltage output that result from a certain displacement. Taking readings from many different displacements then gives data that can be used to approximate the curve of this data so that from the deferential voltage output we can find the corresponding core displacement.

**Objectives of the Laboratory Experiments**

The purpose of this laboratory experiment is mainly calibration of the Linear Variable Differential Transformer. By taking the output voltage readings of the LVDT at several different known displacements we can construct a curve to represent the behavior of the LVDT voltage output during a span of a 30 mm distance. From this approximated curve we can then find the displacement that corresponds to a certain voltage reading that does not exist in the taken data.

**Experimental Setup**

The linear variable differential transformer is a displace measurement device that uses a movable core that is surrounded by three separate coils. The primary coil in the center is connected to a power source and creates a magnetic field. The two secondary coils on either side of the primary coil are connected in parallel but with opposite polarity and have and induced current due to the magnetic field from the primary coil. When the core is in the very center of these coils, the voltage outputs from each secondary coil are equivalent and opposite so the total output of the LVDT is zero. When the core in the middle is displaced the core manipulates the magnetic field of the primary coil which in turn changes the induced current in each secondary coil. As a result there is a non-zero voltage difference reading because the voltage output of each secondary coil no longer cancels the other out. We then can compare how the voltage output compares to a certain displacement. This experiment is set up with a measuring device that pushes a rod which then in turn pushes on the core of the linear variable differential transformer to change the resulting voltage reading.

**Experimental Procedure and Results**

To complete this experiment we start with a displacement of 0 mm and increase this displacement by 1 mm increments taking down the resulting voltage readings for each increment until we reach a final displacement of 30 mm. After this is complete we repeat the process again coming down from 30 mm to 0 mm to confirm accuracy of the readings and to find the largest measurement hysteresis. The results from this experiment are shown in Table 1 below.

Table 1. Displacements and Resulting Voltages

Displacement (mm) |
Result 1 (V) |
Result 2 (V) |
Difference (V) |

0 |
10.7983 |
10.7983 |
0.0000 |

1 |
10.7947 |
10.7965 |
-0.0018 |

2 |
10.7626 |
10.7658 |
-0.0032 |

3 |
10.6968 |
10.7007 |
-0.0039 |

4 |
10.5942 |
10.5981 |
-0.0039 |

5 |
10.4497 |
10.4602 |
-0.0105 |

6 |
10.2777 |
10.2892 |
-0.0115 |

7 |
10.0791 |
10.0975 |
-0.0184 |

8 |
9.8704 |
9.8910 |
-0.0206 |

9 |
9.6580 |
9.6753 |
-0.0173 |

10 |
9.4400 |
9.4601 |
-0.0201 |

11 |
9.2180 |
9.2433 |
-0.0253 |

12 |
8.9951 |
9.0268 |
-0.0317 |

13 |
8.7868 |
8.8129 |
-0.0261 |

14 |
8.5629 |
8.5970 |
-0.0341 |

15 |
8.3550 |
8.3844 |
-0.0294 |

16 |
8.1366 |
8.1743 |
-0.0377 |

17 |
7.9234 |
7.9630 |
-0.0396 |

18 |
7.7097 |
7.7555 |
-0.0458 |

19 |
7.5018 |
7.5447 |
-0.0429 |

20 |
7.2875 |
7.3412 |
-0.0537 |

21 |
7.0769 |
7.1301 |
-0.0532 |

22 |
6.8637 |
6.9184 |
-0.0547 |

23 |
6.6496 |
6.6994 |
-0.0498 |

24 |
6.0940 |
6.1129 |
-0.0189 |

25 |
4.8386 |
4.8421 |
-0.0035 |

26 |
3.5417 |
3.5350 |
0.0067 |

27 |
2.2569 |
2.2440 |
0.0129 |

28 |
0.9830 |
0.9812 |
0.0018 |

29 |
-0.2122 |
-0.2123 |
0.0001 |

30 |
-0.6312 |
-0.6333 |
0.0021 |

**Data Analysis, Interpretation, and Discussion**

The largest value of measurement hysteresis is .0547 V and occurs at a measured displacement is 22 mm.

Using this data we can get a best fit curve to further evaluate the data. A fourth order polynomial trend line produces a curve that is very close to the plotted points, I therefore use this trend line equation as the equation of best curve fit for the data. The equation for the polynomial trend line is:

y=-6E-05x^{4}+.0028x^{3}-.0429x^{2}+.0739x+10.804

Where ‘y’ corresponds to the voltage reading and ‘x’ refers to the distance measured.

We can then use this equation as a basis to estimate the displacement corresponding to a voltage reading of 9.2857 V.

9.2857=-6E-05x^{4}+.0028x^{3}-.0429x^{2}+.0739x+10.804

Solving for x we then get a displacement of __10.9756 mm.__

**Conclusions**

In completion of this particular experiment of calibration of the linear transformer the student gains a basic knowledge of how and why calibrations are performed. We can also see by the visual representations of this data that the relationship between the displacement of the core and the resulting differential voltage output is not a linear one.

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